Complexified Type IIB Supergravity

• Complexified Type IIB supergravity is a framework where all fields are defined as holomorphic sections, unifying standard type IIB, IIB★, and other real forms.
• The formulation uses a detailed Ansatz with complex warp factors and holomorphic functions on a complex base, enforcing F(4) symmetry via Killing spinor reduction.
• Analytic continuation within this framework bridges AdS, dS, and exotic signatures, underpinning holography such as matrix model dualities with broken SO(10) symmetry.

Complexified Type IIB supergravity refers to the extension of the conventional (real) type IIB supergravity theory to a framework in which all bosonic and fermionic fields, and the supersymmetry variations themselves, are complexified. This approach provides a universal structure encompassing all possible real forms of type IIB supergravity—including standard type IIB, type IIB★, and other less common real forms possessing non-standard spacetime signatures or alternative reality properties—and serves as a natural parent space for analyzing solutions with different bosonic symmetry groups, analytic continuations between real geometries of interest (such as AdS, dS, or spheres), and highly symmetric settings relevant to holography, particularly the holography of matrix models with broken SO(10) symmetry.

1. General Construction and Ansatz

The construction of general local solutions to complexified type IIB supergravity invariant under a given complex superalgebra, such as the complexified F(4) superalgebra, begins by considering all fields (metric, fluxes, dilaton, axion, and fermions) as holomorphic sections over a suitable base—typically a complex surface $\Sigma_{\mathbb C}$—with no imposed reality conditions. For F(4)-invariant backgrounds, the complexified geometry is taken as a warped product: $$ds^2 = f_6^2\, ds^2_{\mathcal{M}_{6\mathbb C}} + f_2^2\, ds^2_{\mathcal{M}_{2\mathbb C}} + ds^2_{\Sigma_{\mathbb C}}$$ where $$\mathcal{M}_{6\mathbb C} \cong SO(7;\mathbb C)/SO(6;\mathbb C)$$ is a six-complex-dimensional maximally symmetric space and $$\mathcal{M}_{2\mathbb C} \cong SO(3;\mathbb C)/SO(2;\mathbb C)$$ is a two-complex-dimensional maximally symmetric space (D'Hoker et al., 20 Aug 2025). The warp factors $f_6$ and $f_2$ are holomorphic functions over $\Sigma_{\mathbb C}$, and all fluxes are similarly expressed in a locally holomorphic fashion, such as $H_{(3)} = h_a\, e^a \wedge e^{6 7}$ and analogously for the complexified RR sector.

2. Killing Spinor Reduction and Symmetry Constraints

The imposition of complexified F(4) symmetry translates to invariance under its complex bosonic maximal subalgebra $$\mathfrak{so}(3;\mathbb C) \oplus \mathfrak{so}(7;\mathbb C)$$. Accordingly, the ten-dimensional Killing spinors are decomposed based on this symmetry: the spinor bundle is built as a tensor product over the respective Dirac spinor representation spaces for $\mathcal{M}_{6\mathbb C}$, $\mathcal{M}_{2\mathbb C}$, and $\Sigma_{\mathbb C}$. Projection conditions, such as

$$(\mathbb{I} \otimes \tau^{(11)} \otimes \gamma_{(3)})\, \zeta = \zeta$$

select the desired complex subspace. A set of four complex parameters (say, $\alpha, \beta, \gamma, \delta$) label all supersymmetry-preserving Killing spinors, subject to further projectors and chiralities. The relations among these parameters are directly linked to the holomorphic data on $\Sigma_{\mathbb C}$ and the structure of the warp factors, e.g.,

$$f_6 = c_6(\delta - \beta)\,, \quad f_2 = c_2(\gamma + \alpha)$$

with all functions holomorphic in $\Sigma_{\mathbb C}$ (D'Hoker et al., 20 Aug 2025).

3. Reality Conditions and Classification of Real Forms

Imposing suitable involutive conjugation conditions on the holomorphic solution yields all possible real forms of type IIB supergravity. Acceptable real forms are determined by demanding compatibility with the irreducible real forms of the complexified F(4) superalgebra, whose maximal bosonic subalgebras are exhaustively classified as follows: $$\begin{array}{l} \mathfrak{so}(3) \oplus \mathfrak{so}(1,6)\ \mathfrak{so}(3) \oplus \mathfrak{so}(2,5)\ \mathfrak{so}(1,2) \oplus \mathfrak{so}(7)\ \mathfrak{so}(1,2) \oplus \mathfrak{so}(3,4) \end{array}$$ There is no real form corresponding to a compact $\mathfrak{so}(3) \oplus \mathfrak{so}(7)$, reflecting the non-existence of fully real $S^6 \times S^2 \times \Sigma$ solutions in true real forms of type IIB (with both factors compact), a point of direct relevance to matrix model holography (D'Hoker et al., 20 Aug 2025). Imposing specific conjugation properties on the holomorphic data—e.g., pairing holomorphic functions as complex conjugates—yields standard type IIB (AdS$_6 \times S^2 \times \Sigma$), type IIB$^\star$ (dS$_{1,5} \times S^2 \times \Sigma$), and other forms with exotic signatures.

Table: Example Geometries by Real Form

Real Form	Maximally Symmetric Factors	Signature
Type IIB	$AdS_6 \times S^2 \times \Sigma$	$(- + + + + +)$
Type IIB$^\star$	$dS_{1,5} \times S^2 \times \Sigma$	$(+ - - - - -)$
others	$S^6\times S^2\times \Sigma$ (complex/analytic)	see below

4. Analytic Continuation and the Holography of Polarized Matrix Models

In scenarios where the physically interesting geometry (for example, $S^6 \times S^2 \times \Sigma$) does not arise as a real solution in any of the standard real forms, analytic continuation within the complexified framework becomes essential. For instance, the polarized IKKT matrix model, which admits a deformation breaking SO(10) down to SO(7)×SO(3), is conjectured to be holographically dual to a sector of (complexified) type IIB supergravity carrying F(4) (complex) symmetry. The gravitational dual geometry expected on symmetry and probe grounds is $S^6\times S^2\times \Sigma$. Since this solution does not admit a real section with maximal bosonic symmetry $\mathfrak{so}(7)\oplus\mathfrak{so}(3)$, it is interpreted as a formal analytic continuation from $dS_{1,5}\times S^2\times \Sigma$ or $AdS_6\times S^2\times \Sigma$ geometries realized within type IIB$^\star$ or type IIB, respectively. The complexified solution thus provides the "parent" from which relevant real solutions are obtained via analytic continuation, with physical observables correspondingly analytically continued (D'Hoker et al., 20 Aug 2025).

5. General Mathematical Structure of Solutions

All complexified F(4)-invariant solutions possess a metric and flux Ansatz of the schematic form: $$\begin{align*} ds^2 &= f_6^2\, ds^2_{\mathcal{M}_{6\mathbb C}} + f_2^2\, ds^2_{\mathcal{M}_{2\mathbb C}} + ds^2_{\Sigma_{\mathbb C}}\ H_{(3)} &= h_a\, e^a \wedge e^{67}, \quad \widetilde{F}_{(3)} = g_a\, e^a \wedge e^{67} \end{align*}$$ All field content is determined in terms of underlying holomorphic data (functions of $\Sigma_{\mathbb C}$), with warp factors and flux profiles algebraically and differentially constrained by the reduced Killing spinor equations and Bianchi identities.

Consistent conjugation conditions correspond to the four real forms of F(4), and the explicit choice selects which components of the holomorphic data become complex conjugate pairs, which in turn dictates the detailed signature and bosonic symmetry of the resulting solution (see classification above).

6. Physical Significance and Applications

The complexified type IIB solutions thus obtained serve several roles:

• As universal parent solutions that encode in a single framework all possible real forms and the analytic continuations between them.
• In holography, supporting proposals for duals to deformed (polarized) matrix models such as the SO(7)×SO(3)-invariant IKKT model, where the absence of a suitable real solution necessitates analytic continuation. The correspondence between F(4;\,$\mathbb C$)-invariant supergravity solutions and matrix model vacua is established by matching symmetries and asymptotic structures.
• For classification of local and global real solutions in type IIB, type IIB$^\star$, and related real forms, with direct mapping to previously known AdS$_6$, dS$_6$, and S$^6$ backgrounds (after analytic continuation).

7. Relation to Earlier Real and Complexified Supergravities

Complexification is not merely a formal device but an essential step for:

• Systematic analytic continuation between de Sitter and anti-de Sitter solutions, providing a bridge between real supergravities relevant for holography and those needed for models with positive cosmological constant or exotic signature.
• Understanding the universality of underlying symmetry—the F(4) superalgebra—which admits only four real forms, each corresponding to physically distinct spacetime signatures and geometries.

The approach further clarifies why compact solutions with maximal symmetry SO(7)×SO(3) (i.e. $S^6\times S^2$ with both compact factors) cannot exist as real solutions in type IIB or its real forms, a conclusion with implications for both mathematical classification and physical model building (D'Hoker et al., 20 Aug 2025).

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Summary Table: Key Aspects of Complexified Type IIB Supergravity (F(4) Invariant Solutions)

Aspect	Description	Reference/Formula
Geometry	$$\mathcal{M}_{6\mathbb C} \times \mathcal{M}_{2\mathbb C} \times \Sigma_{\mathbb C}$$	$$ds^2= f_6^2 ds^2_{\mathcal{M}_{6\mathbb C}} + f_2^2 ds^2_{\mathcal{M}_{2\mathbb C}} + ds^2_{\Sigma_{\mathbb C}}$$
Fields	All bosonic/fermionic fields holomorphic; fluxes via $H_{(3)}$, $\widetilde{F}_{(3)}$	$H_{(3)} = h_a e^a \wedge e^{67}$ etc.
Reality Forms	All possible by choice of involution; classified by F(4) real forms	see maximal bosonic subalgebras above
Physical Applications	Unification of AdS/dS/sphere backgrounds; polarized IKKT model holography	Analytic continuation enables $S^6\times S^2\times\Sigma$
Analytic Continuation	Enables passing between real forms/geometries missing compact solutions in real IIB

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In conclusion, the complexified type IIB supergravity framework gives a comprehensive, symmetry-based, and holomorphic parameterization of solutions admitting F(4) invariance, encodes all real solutions as well as analytic continuations between them, and underpins the holographic dual description of matrix models with SO(7)×SO(3) symmetry in the absence of strictly real-geometric duals (D'Hoker et al., 20 Aug 2025).